This paper establishes the Theory of Curved Space (TCS) as a complete geometric and operational alternative to the Minkowski spacetime paradigm. TCS is fundamentally grounded in a four-dimensional Euclidean ontology (E⁴) with absolute time. We rigorously demonstrate that the kinematic and dynamic structures of Special Relativity (STR) are not fundamental properties of spacetime, but operational artifacts arising exclusively from the closure of a radar measurement loop performed by observers situated on oblique 3D slices of E⁴. At the ontological level, an inertial frame is defined as a 3D slice inclined at an angle (= = v/c) in E⁴, which induces a purely geometric transverse projection contraction (1/). We prove that the observed light front of a moving source is not a section of the fundamental 4D sphere, but an affine ellipsoidal image of the source's proper 3D sphere. From the focal asymmetry of this ellipsoid, characterized by longitudinal branches R_ = R₀ (1), we extract the elementary branch factor of the closed radar loop: = R_+{R_-} = 1+{1-} = eS, S = artanh. axiomatizing the diagonal action of the closed loop in the light basis, ₋₂ = diag (D, D^-1), we derive the exact Lorentz transformation matrix in the standard basis (ct, x). The Minkowski interval c²t² - x² emerges not as a fundamental metric, but as the product of two conjugate light branches invariant under ₋₂ = 1. Consequently, the relativity of simultaneity and time dilation are reinterpreted as secondary operational effects of phase closure, preserving a strictly Euclidean ontology with absolute time. The dynamic sector is extended by introducing an internal projection mask M, reflecting the local deformation of the 3D slice, which generalizes the dispersion relation to: ² = p²c² + m²c⁴ M². identify M = eq, where q is the isotropic diagonal element of the deformation tetrad field Q (x, t) SL (4, R). The fundamental dynamic postulate of TCS is the Conservation of Homothety (Q = 1), establishing the preservation of the 4D volume as the Parent Conservation Law. All standard conservations (energy, momentum, charge) are derived as 3D projections of this single geometric invariant. Mass is reinterpreted as an oscillation of geometry along the hidden 4th axis with Compton amplitude, naturally yielding E=mc² = h. Gravity arises not as spacetime curvature, but as the elastic response of the 4D vacuum, characterized by a stiffness modulus K, which is calibrated by the Newtonian limit to yield G = c⁴ / (2K). This elastic paradigm intrinsically eliminates spacetime singularities and provides a physical mechanism for the weakness of gravity. Furthermore, we prove a canonical mapping of the configuration space of Q into effective internal symmetries. Via the polar decomposition of the 3D block Q₃ = RS and the construction of the Hermitian matrix H = X + i (where X = S and = R), we generate a composite unitary image (Q₃) = (iH) SU (3). Fixing a complex line in C³ canonically reduces this sector to SU (2) U (1). The mixed components of Q form a decoupled hidden sector. Physical Perspectives and Interpretations: The TCS framework provides unified geometric solutions to the deepest open problems in fundamental physics: - Spin 1/2: Arises as the 4D rotation (somersault) of massive solitons around the x₄ axis, explaining its exclusive association with massive particles and its 720^ periodicity as a double loop in the hidden dimension. - Strong Confinement: Modeled via the topology of a local 4D tesseract. Quarks are 3D faces of this structure; their isolation is topologically forbidden by the Homothety postulate (Q = 1), requiring infinite energy to sever a 3D face from its 4D volume. - Weak Chirality: The SU (2) L structure is interpreted as the geometric necessity of a specific rotational direction (the "somersault") for the tesseract in E⁴, naturally breaking parity. - Vacuum Energy Catastrophe: The 120-orders-of-magnitude discrepancy is resolved, as the global Homothety strictly enforces a zero fundamental 4D vacuum energy; observed dark energy is merely the centrifugal effect of global 4D rotation. - Quantum Uncertainty: The Heisenberg principle is derived as a direct consequence of information loss during the projection from continuous E⁴ to the operational 3D slice, rendering the universe fundamentally continuous but discretely observed. - Cosmological Tensions: Discrepancies such as the Hubble tension naturally emerge from the variance of the observational mask M (t) across different cosmological epochs. Thus, TCS constructs a singularity-free, unified Euclidean alternative to the pseudo-Riemannian paradigm, seamlessly bridging operational kinematics, elastic gravity, and the phenomenology of quantum interactions within a single geometric ontology based on E⁴ and absolute time.
Говорушкин et al. (Sat,) studied this question.